Hamilton-Jacobi Theory and the Quantum Action Variable

Leacock, Robert A.; Padgett, Michael J.

doi:10.1103/physrevlett.50.3

Cited by 143 publications

(176 citation statements)

References 9 publications

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“…(4). It shall be noted that ρ is defined only over that region of the x r x i -plane, where characteristic curves which cross the real axis pass.…”

Section: Discussionmentioning

confidence: 99%

“…(1) was used by many physicists such as Wentzel, Pauli and Dirac, even during the time of inception of quantum mechanics [2]. In a commendable work in 1982, Leacock and Padget [4] have used the QHJE to obtain eigenvalues in many bound state problems, without actually having to solve the corresponding Schrodinger equation. However, there were no trajectories in their work and it was only in [1] that the equation of motion (2) explicitly solved and the complex trajectories of particles in any quantum state obtained and drawn, for the first time.…”

Section: Introductionmentioning

confidence: 99%

“…In a previous work [1], the quantum Hamilton-Jacobi equation [2][3][4][5] (QHJE) was made use of to demonstrate the existence of particle trajectories in a complex space, for different quantum states. This complex quantum trajectory representation was obtained by modifying the de Broglie-Bohm (dBB) approach to quantum mechanics [6], which allows particle motion guided by the wave function.…”

Section: Introductionmentioning

confidence: 99%

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Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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It is shown that in the complex trajectory representation of quantum mechanics, the Born's Ψ ⋆ Ψ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

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“…(4). It shall be noted that ρ is defined only over that region of the x r x i -plane, where characteristic curves which cross the real axis pass.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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“…The QHJ formalism, formulated as a theory analogous to the classical canonical transformation theory [13], [14], [15], was proposed by Leacock and Padgett in 1983. It has been applied to one dimensional bound state problems and separable problems in higher dimensions [11].…”

Section: Quantum Hamilton -Jacobi Formalismmentioning

confidence: 99%

“…We then use the quantum Hamilton-Jacobi (QHJ) approach [11], which naturally takes advantage of the singularities of the new potential, to isolate the domains corresponding to discrete and band spectra. The subtle aspects of the boundary conditions in quantum mechanics, which lead to the existence of both bound states and band structure in the Scarf potential, come out naturally in this approach.…”

Section: Introductionmentioning

confidence: 99%

Bound states and band structure—A unified treatment through the quantum Hamilton–Jacobi approach

2005

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We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.

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Regularity and profiles of solutions to a higher order Eyring–Powell fluid with Darcy–Forchheimer porosity term

Rahman

Palencia

2022

Math Methods in App Sciences

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A flow of Eyring–Powell type constitutes a remarkable area of analysis to model non‐Newtonian processes in fluids. The associated diffusion term comes from the general kinetic theory of liquids and permits to account for a wider diffusivity, which is applicable for qualitatively low to higher shear stresses. The goal of the present article is to introduce a generalization of an Eyring–Powell fluid by the introduction of a porous reaction term (of Darcy–Forchheimer type) and a perturbation with a higher order operator. In particular, we consider that our model is an extension of a classical Eyring–Powell fluid in the same manner as introduced for other equations (see the extended Fisher–Kolmogorov model). The obtained equation is novel and requires analysis about existence, regularity and uniqueness of solutions. Stationary solutions are explored under the definition of a Hamiltonian. In addition, profiles of solutions are obtained with an exponential scaling that ends in a Hamilton–Jacobi equation. Eventually, some numerical assessments are introduced to validate the hypothesis done, and to discuss about the accuracy of the analytical approach followed.

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Hamilton-Jacobi Theory and the Quantum Action Variable

Cited by 143 publications

References 9 publications

Probability and complex quantum trajectories

Probability and complex quantum trajectories

Bound states and band structure—A unified treatment through the quantum Hamilton–Jacobi approach

Regularity and profiles of solutions to a higher order Eyring–Powell fluid with Darcy–Forchheimer porosity term

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